Injectivity and modelling in the blass topos

Abstract

A very obvious general question concerning the modelling of any familiar mathematical notion cfJ, with associated relevant maps, in some topos E, is how properties of the 'usual' category determined by e:p relate to the properties of the corresponding category of cfJ-objects in E. This paper is an illustration of what can happen in this context with regard to injectivity in the categories concerned. Blass [4,6] showed, for a certain topos B, that the category of abelian groups in B has no non-zero injectives although the 'usual' category of abelian groups enjoys the familiar property of having enough injectives, meaning that any of its objects can be embedded into an injective one. We refer to this occurrence as a breakdown of injectivity (in B). In the following, we exhibit further such breakdowns, using the same topos as Blass [4] and appropriate variants of his method of proof. The notions we deal with are: modules over any ring, Boolean algebras, and distributive lattices. Regarding the latter two, it might be recalled that, in the usual setting, the injective Boolean algebras are exactly the complete ones [12, p. 112] and the injective distributive lattices the complete Boolean algebras [3]. The existence of enough injectives then follows fairly easily in either case. By way of contrast, we also discuss three notions for which the existence of enough injectives is retained when they are modelled in the topos considered here, thus showing that the breakdown of injectivity in the earlier situations is by no means a general feature of that topos. The three concepts of this second type are partially ordered sets, semilattices, and normed linear spaces, the maps being, respectively, the order preserving maps, the homomorphisms, and the linear contractions; here, injectivity is understood with respect to embeddings (rather than mere monomorphisms) in the first and third case. We note that, in the usual situation, the injectives are, respectively, the complete partially ordered sets [2], the locales, i.e. the complete partially ordered sets in which the distribution law X/ V xa '= V (xvxJ holds for binary meets (1\) and arbitrary joins (V) [7], and the